Question

The locus of centre of the circle which touches the circle $x^2+(y-1{)}^2=1$ externally and also touches x-axis is

• A. $(x,y):x^2+(y-1{)}^2=4\cup (x,y):y>0$
• B. $(x,y):x^2=4y\cup (0,y):(y<0)$
• C. $(x,y):x^{2}y\cup (0,y):y<0$
• D. $(x,y):x^2=4y\cup (x,y):y<0$

Answer 1

Answer: (B)

$(x,y):x^2=4y\cup (0,y):(y<0)$

As per given $C_1{C}_2=r_1+r_2$
$\therefore$ $\sqrt{(h-0{)}^2+(k-1{)}^2}=1+|k|$
$\Rightarrow$ $h^2+(k-1{)}^2=(1+|k|{)}^2$
$\Rightarrow$ $h^2=2k+2|k|$. Hence, locus is $x^2-2y=2|y|$
$y>0,x^2=4y$ and $y<0,x^2=0$ ,$\Rightarrow$ $x=0$